{"title":"An Optimal Lower Bound for Smooth Convex Functions","authors":"Mihai I. Florea, Yurii E. Nesterov","doi":"10.1007/s10208-025-09712-y","DOIUrl":"https://doi.org/10.1007/s10208-025-09712-y","url":null,"abstract":"<p>First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds leads to an increase in theoretical guarantees and in observed practical performance. In this work, we define a global lower bound for smooth objectives that is optimal with respect to the collected oracle information. Our bound can be readily employed by the Gradient Method with Memory to improve its performance. Further using the machinery underlying the optimal bounds, we introduce a modified version of the estimate sequence that we use to construct an Optimized Gradient Method with Memory possessing the best known convergence guarantees for its class of algorithms up to the proportionality constant. We additionally equip the method with an adaptive convergence guarantee adjustment procedure that is an effective replacement for line-search. Simulation results on synthetic but otherwise difficult smooth problems validate the theoretical properties of the bound and of the proposed methods.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Raymond van Bommel, Jordan Docking, Vladimir Dokchitser, Reynald Lercier, Elisa Lorenzo García
{"title":"Reduction of Plane Quartics and Cayley Octads","authors":"Raymond van Bommel, Jordan Docking, Vladimir Dokchitser, Reynald Lercier, Elisa Lorenzo García","doi":"10.1007/s10208-025-09704-y","DOIUrl":"https://doi.org/10.1007/s10208-025-09704-y","url":null,"abstract":"<p>We give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in <i>p</i>-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of “cluster pictures” for hyperelliptic curves. We also construct explicit families of quartic curves that realise all possible stable types, against which we test these criteria. We give numerical examples that illustrate how to use these criteria in practice.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"27 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs","authors":"Chupeng Ma","doi":"10.1007/s10208-025-09711-z","DOIUrl":"https://doi.org/10.1007/s10208-025-09711-z","url":null,"abstract":"<p>Multiscale partial differential equations (PDEs), featuring heterogeneous coefficients oscillating across possibly non-separated scales, pose computational challenges for standard numerical techniques. Over the past two decades, a range of specialized methods has emerged that enables the efficient solution of such problems. Two prominent approaches are numerical multiscale methods with problem-adapted coarse approximation spaces, and structured inverse methods that exploit a low-rank property of the associated Green’s functions to obtain approximate matrix factorizations. This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in Babuska and Lipton (Multiscale Model Simul 9:373–406, 2011). MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to the number of local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with <span>(L^{infty })</span>-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems and higher-order problems. Notably, we prove a local convergence rate of <span>(O(e^{-cn^{1/d}}))</span> for MS-GFEM for all these problems, improving upon the <span>(O(e^{-cn^{1/(d+1)}}))</span> rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green’s functions admit an <span>(O(|log epsilon |^{d}))</span>-term separable approximation on well-separated domains with error <span>(epsilon >0)</span>. Our analysis improves and generalizes the result in Bebendorf and Hackbusch (Numerische Mathematik 95:1–28, 2003) where an <span>(O(|log epsilon |^{d+1}))</span>-term separable approximation was proved for Poisson-type problems. It provides a rigorous theoretical foundation for diverse structured inverse methods, and also clarifies the intimate connection between approximation mechanisms in such methods and MS-GFEM.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"45 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143889834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Symbolic Summation of Multivariate Rational Functions","authors":"Shaoshi Chen, Lixin Du, Hanqian Fang","doi":"10.1007/s10208-025-09710-0","DOIUrl":"https://doi.org/10.1007/s10208-025-09710-0","url":null,"abstract":"<p>Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato’s isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard’s problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"15 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143872731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local Geometry Determines Global Landscape in Low-Rank Factorization for Synchronization","authors":"Shuyang Ling","doi":"10.1007/s10208-025-09707-9","DOIUrl":"https://doi.org/10.1007/s10208-025-09707-9","url":null,"abstract":"<p>The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, <span>(mathbb {Z}_2)</span>-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer–Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the rank of the factorization exceeds twice the condition number of the “Laplacian\" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer–Monteiro factorization is robust to “monotone adversaries\", mirroring the resilience of the SDR. In other words, introducing “favorable\" adversaries into the data will not result in the emergence of new spurious local minimizers.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"2 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143872732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Adaptive Mesh Refinement for Arbitrary Initial Triangulations","authors":"Lars Diening, Lukas Gehring, Johannes Storn","doi":"10.1007/s10208-025-09698-7","DOIUrl":"https://doi.org/10.1007/s10208-025-09698-7","url":null,"abstract":"<p>We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes. Our ansatz allows for the intrinsic use of existing implementations.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"21 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local Space-Preserving Decompositions for the Bubble Transform","authors":"Richard Falk, Ragnar Winther","doi":"10.1007/s10208-025-09700-2","DOIUrl":"https://doi.org/10.1007/s10208-025-09700-2","url":null,"abstract":"<p>The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in <span>(L^2)</span>, and satisfies the <i>stable decomposition property</i>.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"61 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143641098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Towards a Fluid Computer","authors":"Robert Cardona, Eva Miranda, Daniel Peralta-Salas","doi":"10.1007/s10208-025-09699-6","DOIUrl":"https://doi.org/10.1007/s10208-025-09699-6","url":null,"abstract":"<p>In 1991, Moore (Nonlinearity 4:199–230, 1991) raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao (J Am Math Soc 29(3):601–674, 2016) asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in Cardona et al. (Proc Natl Acad Sci 118(19):e2026818118, 2021) of a “Fluid computer” in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense (Chern and Hamilton in On Riemannian metrics adapted to three-dimensional contact manifolds, Springer, Berlin, 1985). We also sketch the completely different construction for the Euclidean metric in <span>(mathbb {R}^3)</span> as given in Cardona et al. (J Math Pures Appl 169:50–81, 2023). These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"16 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence","authors":"Riccardo Bonalli, Alessandro Rudi","doi":"10.1007/s10208-025-09705-x","DOIUrl":"https://doi.org/10.1007/s10208-025-09705-x","url":null,"abstract":"<p>We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"11 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143569769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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